10 7, x + 30x + 36 SOLUTION: 89 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.


 Garey Eaton
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1 Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect square. 36 = 6 The middle term is equal to ab. 60x = (5x)(6) So, 5x + 60x + 36 is a perfect square trinomial..6x + 30x + 36 SOLUTION: The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 3.x x 8 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In this trinomial, a =, b = 1 and c = 8, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of ( 8) or 56 with a sum of 1. Factors of 56 Sum 1, , 56 55, 8 6, 8 6 4, , , 8 1 7, 8 1 The correct factors are 7 and 8. Page 1
2 .6x + 30x + 36 SOLUTION: Squares The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 3.x x 8 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In this trinomial, a =, b = 1 and c = 8, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of ( 8) or 56 with a sum of 1. Factors of 56 Sum 1, , 56 55, 8 6, 8 6 4, , , 8 1 7, 8 1 The correct factors are 7 and x 34x + 48 SOLUTION: Factor the GCF of from each term. The resulting polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 3, b = 17 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 3(4) or 60 with a sum of 17. Factors of 7 Sum 1, 7 73, , 4 7 4, 18 6, , 9 17 The correct factors are 8 and 9. Page
3 Squares 4.6x 34x + 48 SOLUTION: Factor the GCF of from each term. The resulting polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 3, b = 17 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 3(4) or 60 with a sum of 17. Factors of 7 Sum 1, 7 73, , 4 7 4, 18 6, , 9 17 The correct factors are 8 and x + 64 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor the GCF. The greatest common factor of each term is x + 9x 16 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 4, b = 9 and c = 16, so m + p is positive and mp is negative. Therefore, m and p must have differentsigns. Listthefactorsof4( 16) or 64 with a sum of 9. Factors of 64 Sum 1, esolutions Manual Powered by Cognero63 1, 64, 3 30, 3 30 Page 3
4 The greatest common factor of each term is 4. Squares 6.4x + 9x 16 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 4, b = 9 and c = 16, so m + p is positive and mp is negative. Therefore, m and p must have differentsigns. Listthefactorsof4( 16) or 64 with a sum of 9. Factors of 64 1, 64 1, 64, 3, 3 4, 16 4, 16 4, 15 8, 8 Sum Therearenofactorsof 64withasumof9. So, this trinomial is prime. Solve each equation. Confirm your answers using a graphing calculator. 7.4x = 36 SOLUTION: The roots are 3 and 3. Confirm the roots using a graphing calculator. Let Y1 = 4x and Y = 36. Use the intersect option from the CALC menutofindthepointsofintersection. [ 5, 5] scl: 1 by [ 5, 45] scl: 5 [ 5, 5] scl: 1 by [ 5, 45] scl: 5 Thus,thesolutionsare 3 and a 40a = 16 SOLUTION: Rewrite with 0 on the right side. esolutions Manual  Powered by Cognero Page 4
5 [ 5, 5] scl: 1 by [ 5, 45] scl: 5 [ 5, 5] scl: 1 by [ 5, 45] scl: 5 Squares Thus,thesolutionsare 3 and a 40a = 16 SOLUTION: Rewrite with 0 on the right side. Thetrinomialisaperfectsquaretrinomial. The root is or0.8. Confirm the root using a graphing calculator. Let Y1 = 5a 40a and Y = 16. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 5] scl: 1 by [ 5, 5] scl: 3 Thus the solution is. 9.64y 48y + 18 = 9 SOLUTION: Rewrite the trinomial with  on the right side. Page 5
6 [ 5, 5] scl: 1 by [ 5, 5] scl: 3 Thus thesquares solution is. 9.64y 48y + 18 = 9 SOLUTION: Rewrite the trinomial with  on the right side. The resulting trinomial is a perfect square trinomial. The root is or Confirm the roots using a graphing calculator. Let Y1 = 64y 48y + 18 and Y = 9. Use the intersect option from the CALCmenutofindthepointsofintersection. [.5,.5] scl: 0.5 by [0, 0] scl: Thus, the solution is. 10.(z + 5) = 47 SOLUTION: Page 6 The roots are and or about 11.86and1.86.
7 [.5,.5] scl: 0.5 by [0, 0] scl: Thus, thesquares solution is. 10.(z + 5) = 47 SOLUTION: The roots are and or about 11.86and1.86. Confirm the roots using a graphing calculator. Let Y1 = (z + 5) and Y = 47. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 15, 5] scl: 1 by [ 5, 55] scl: 3 [ 15, 5] scl: 1 by [ 5, 55] scl: 3 and Thus, the solutions are or about and CCSSREASONING While painting his bedroom, Nick drops his paintbrush off his ladder from a height of 6 feet. Use the formula h = 16t + h 0 to approximate the number of seconds it takes for the paintbrush to hit the floor. SOLUTION: Let h = 0 feet and h 0 = 6 feet. The roots are 0.6 and 0.6. The time the paint brush falls cannot be negative. So, it takes about 0.6 second for the paintbrush to hit the floor. Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.4x 4x SOLUTION: The last term is not a perfect square. So, 4x 4x is not a perfect square trinomial. esolutionsmanual  Powered by Cognero 13.16x 56x + 49 SOLUTION: Page 7
8 1.4x 4x SOLUTION: Squares The last term is not a perfect square. So, 4x 4x is not a perfect square trinomial x 56x + 49 SOLUTION: The first term is a perfect square. 16x = (4x) The last term is a perfect square. 49 = (7) The middle term is equal to ab. 56x = (4x)(7) So, 16x 56x + 49 is a perfect square trinomial x 90x + 5 SOLUTION: The first term is a perfect square. 81x = (9x) The last term is a perfect square. 5 = (5) The middle term is equal to ab. 90x = (9x)(5) So, 81x 90x + 5 is a perfect square trinomial. 15.x + 6x SOLUTION: The last term is not a perfect square. So, x + 6x is not a perfect square trinomial. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 16.4d Manual + 39d Powered 18 by Cognero esolutions SOLUTION: Factor GCF of 3 from each term. Page 8
9 15.x + 6x SOLUTION: Squares The last term is not a perfect square. So, x + 6x is not a perfect square trinomial. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 16.4d + 39d 18 SOLUTION: Factor GCF of 3 from each term. The resulting polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the factored trinomial, a = 8, b = 13 and c = 6, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 8( 6) or 48 with a sum of 13. Factors of 48 1, 48 1, 48, 5, 5 3, 16 3, 16 4, 1 4, 1 6, 8 6, 8 Sum The correct factors are 3 and x + 10x 1 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 8, b = 10 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 8( 1) or 168 with a sum of 10. Factors of 60 Sum 1, , , 84 8, , Page 9
10 Squares 17.8x + 10x 1 SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, a = 8, b = 10 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 8( 1) or 168 with a sum of 10. Factors of 60 1, 168 1, 168, 84, 84 3, 56 3, 56 4, 4 4, 4 6, 8 6, 8 8, 1 8, 1 1, 14 1, 14 Sum There are no factors of 8( 1) or 168withasumof10. So, this trinomial is prime. 18.b + 1b 4 SOLUTION: The greatest common factor of each term is. b + 1b 4 = (b + 6b 1) The resulting polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, b = 6 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 1 with a sum of 6. Sum Factors of , 1 1, , 6, , 4 1 3, 4 Therearenofactorsof 1 with a sum of 6. Thus b + 6b 1isprime. Therefore, (b + 6b 1)isfactoredformform. Page 10
11 1, 14 There are no factors of 8( 1) or 168withasumof10. Squares So, this trinomial is prime. 18.b + 1b 4 SOLUTION: The greatest common factor of each term is. b + 1b 4 = (b + 6b 1) The resulting polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the trinomial, b = 6 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 1 with a sum of 6. Sum Factors of , 1 1, , 6, 6 4 3, , 4 Therearenofactorsof 1 with a sum of 6. Thus b + 6b 1isprime. Therefore, (b + 6b 1)isfactoredformform. 19.8y 00z SOLUTION: Factor out the common factor 8. isadifferenceofsquare. Then 0.16a 11b SOLUTION: Thepolynomialisdifferenceofsquares m m 70m SOLUTION: Page 11
12 Squares 3 1.1m m 70m SOLUTION: The polynomial is not a perfect square or a difference of squares. Try to factor using the general factoring pattern. In the factored trinomial, a = 6, b = 11 and c = 35, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 6( 35) or 10 with a sum of 11. Factors of 10 1, 10 1, 10, 105, 105 3, 70 3, 70 5, 4 5, 4 6, 35 6, 35 7, 30 7, 30 10, 1 10, 1 Sum The correct factors are 10 and 1..8c 88c + 4 SOLUTION: After factoring out, the polynomial is a perfect square. Page 1
13 Squares.8c 88c + 4 SOLUTION: After factoring out, the polynomial is a perfect square. 3.1x 84x SOLUTION: After factoring out a 3, the polynomial is a perfect square. 4 4.w w SOLUTION: Thepolynomialisadifferenceofsquares p 3p SOLUTION: After factoring out 3p, the polynomial is a difference of squares q 48q + 36q SOLUTION: After factoring out 4q,thepolynomialisaperfectsquare. Page 13
14 Squares q 48q + 36q SOLUTION: After factoring out 4q,thepolynomialisaperfectsquare t + 10t 84t SOLUTION: Factor out the GCF. Thepolynomialisnotaperfectsquareoradifferenceofsquares. Try to factor using the general factoring pattern. In this trinomial, a =, b = 5 and c = 4, so m + p is positive and mpisnegative. List the positive and negative factors of ( 4) or 84, and identify the factors which sum to 5. Factors of 84 1, 84 1, 84, 4, 4 3, 8 3, 8 4, 14, 1 6, 14 6, 14 7, 1 7, 1 Sum The correct factors are 1 and 7. Page 14
15 Squares 3 7.4t + 10t 84t SOLUTION: Factor out the GCF. Thepolynomialisnotaperfectsquareoradifferenceofsquares. Try to factor using the general factoring pattern. In this trinomial, a =, b = 5 and c = 4, so m + p is positive and mpisnegative. List the positive and negative factors of ( 4) or 84, and identify the factors which sum to 5. Factors of 84 1, 84 1, 84, 4, 4 3, 8 3, 8 4, 14, 1 6, 14 6, 14 7, 1 7, 1 Sum The correct factors are 1 and x + x y 4x 8y SOLUTION: There are four terms, so factor by grouping. Page 15
16 Squares 3 8.x + x y 4x 8y SOLUTION: There are four terms, so factor by grouping. 3 9.a b a ab + ab SOLUTION: There are four terms, so factor by grouping r r 7r + 36 SOLUTION: There are four terms, so factor by grouping k 4k + 48k SOLUTION: After factoring 3k, the polynomial is a perfect square trinomial. Page 16
17 Squares k 4k + 48k SOLUTION: After factoring 3k, the polynomial is a perfect square trinomial c d 10c d + 4c d 10cd SOLUTION: There are four terms, so factor by grouping. 33.g + g 3h + 4h SOLUTION: The GCF of the terms g, g, 3h, and 4h is 1, so there is no GCF to factor out. Since there are four terms, consider factor by grouping. Only the pairs of the first two terms and the last two terms have GCFs other than 1, so try factoring using this grouping. There is no common binomial factor, so this polynomial cannot be written as a product. Thus the polynomial g + g 3h + 4hcannotbefactored. It is prime. Solve each equation. Confirm your answers using a graphing calculator. esolutions 34.4m Manual 4m+Powered 36 = 0by Cognero SOLUTION: Page 17
18 There is no common binomial factor, so this polynomial cannot be written as a product. Thus the polynomial g + g 3h + 4hcannotbefactored. Squares It is prime. Solve each equation. Confirm your answers using a graphing calculator. 34.4m 4m + 36 = 0 SOLUTION: The GCF of the terms is 4, so factor it out. Therootis3. Confirm the root using a graphing calculator. Let Y1 = 4m 4m + 36 and Y = 0. Use the intersect option from the CALCmenutofindthepointsofintersection. [, 8] scl: 1 by [ 5, 5] scl: 1 Thus, the solution is (y 4) = 7 SOLUTION: Therootsare and orabout1.35and6.65. Confirm the roots using a graphing calculator. Let Y1 = (y 4) and Y = 7. Use the intersect option from the CALCmenutofindthepointsofintersection. Page 18
19 [, 8] scl: 1 by [ 5, 5] scl: 1 Thus, thesquares solution is (y 4) = 7 SOLUTION: Therootsare and orabout1.35and6.65. Confirm the roots using a graphing calculator. Let Y1 = (y 4) and Y = 7. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 10] scl: 1 by [ 5, 10] scl: 1 Thus the solutions are and [ 5, 10] scl: 1 by [ 5, 10] scl: SOLUTION: TheGCFofthetermsis1.Thepolynomial is a perfect square trinomial. The root is or about Confirm the roots using a graphing calculator. Let Y1 = the CALCmenutofindthepointsofintersection. and Y = 0. Use the intersect option from Page 19
20 [ 5, 10] scl: 1 by [ 5, 10] scl: 1 Squares Thus the solutions are and [ 5, 10] scl: 1 by [ 5, 10] scl: SOLUTION: TheGCFofthetermsis1.Thepolynomial is a perfect square trinomial. The root is or about Confirm the roots using a graphing calculator. Let Y1 = the CALCmenutofindthepointsofintersection. and Y = 0. Use the intersect option from [ 5, 5] scl: 1 by [ 5, 5] scl: 1 Thus, the solution is. 37. SOLUTION: The GCF of the terms is 1. The polynomial is a perfect square trinomial. Page 0
21 [ 5, 5] scl: 1 by [ 5, 5] scl: 1 Thus, thesquares solution is. 37. SOLUTION: The GCF of the terms is 1. The polynomial is a perfect square trinomial. The root is or Confirm the roots using a graphing calculator. Let Y1 = the CALCmenutofindthepointsofintersection. and Y = 0. Use the intersect option from [ 5, 5] scl: 1 by [ 5, 5] scl: 1 Thus, the solution is or x + 8x + 16 = 5 SOLUTION: The GCF of the terms is 1. The polynomial x + 8x+16isaperfectsquaretrinomial. Page 1
22 [ 5, 5] scl: 1 by [ 5, 5] scl: 1 Thus, thesquares solution is or x + 8x + 16 = 5 SOLUTION: The GCF of the terms is 1. The polynomial x + 8x+16isaperfectsquaretrinomial. The roots are 9and1. Confirm the roots using a graphing calculator. Let Y1 = x + 8x + 16 and Y = 5. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 15, 5] scl: by [ 5, 35] scl: 4 [ 15, 5] scl: by [ 5, 35] scl: 4 Thus, the solutions are 9 and x 60x = 180 SOLUTION: Writewith0ontherightsideandfactoroutGCFof5. Therootis6. Page
23 [ 15, 5] scl: by [ 5, 35] scl: 4 Squares Thus, the solutions are 9 and 1. [ 15, 5] scl: by [ 5, 35] scl: x 60x = 180 SOLUTION: Writewith0ontherightsideandfactoroutGCFof5. Therootis6. Confirm the root using a graphing calculator. Let Y1 = 5x 60x and Y = 180. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 15] scl: 1 by [ 00, 10] scl: 0 Thus, the solution is x = 80x 400 SOLUTION: Rewritethetrinomialwith0ontherightside.TheGCFofthetermsis4,sofactoritout. Page 3
24 [ 5, 15] scl: 1 by [ 00, 10] scl: 0 Squares Thus, the solution is x = 80x 400 SOLUTION: Rewritethetrinomialwith0ontherightside.TheGCFofthetermsis4,sofactoritout. Therootis10. Confirm the roots using a graphing calculator. Let Y1 = 4x and Y = 80x 400. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 5] scl: 1 by [ 450, 550] scl: 150 Thus, the solution is x = 81x SOLUTION: Rewritethetrinomialwith0ontherightside.TheGCFofthetermsis9,sofactoritout. Page 4
25 [ 5, 5] scl: 1 by [ 450, 550] scl: 150 Squares Thus, the solution is x = 81x SOLUTION: Rewritethetrinomialwith0ontherightside.TheGCFofthetermsis9,sofactoritout. The root is orabout0.33. Confirm the roots using a graphing calculator. Let Y1 = 9 54x and Y = 81x. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 1, 1] scl: 0.5 by [ 0, 0] scl: 5 Thus, the solution is. 4.4c + 4c + 1 = 15 SOLUTION: The GCF of the terms is 1. The polynomial 4c + 4c+1isaperfectsquaretrinomial. Page 5
26 [ 1, 1] scl: 0.5 by [ 0, 0] scl: 5 Thus, thesquares solution is. 4.4c + 4c + 1 = 15 SOLUTION: The GCF of the terms is 1. The polynomial 4c + 4c+1isaperfectsquaretrinomial. and The roots are or about.44and1.44. Confirm the roots using a graphing calculator. Let Y1 = 4c + 4c + 1 and Y = 15. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 5] scl: 1 by [ 10, 0] scl: 3 [ 5, 5] scl: 1 by [ 10, 0] scl: 3 Thus, the solutions are and. 43.x 16x + 64 = 6 SOLUTION: The GCF of the terms is 1. The polynomial x 16x + 64 is a perfect square trinomial. Page 6 The roots are and or5.55and10.45.
27 [ 5, 5] scl: 1 by [ 10, 0] scl: 3 [ 5, 5] scl: 1 by [ 10, 0] scl: 3 Thus, thesquares solutions are and. 43.x 16x + 64 = 6 SOLUTION: The GCF of the terms is 1. The polynomial x 16x + 64 is a perfect square trinomial. The roots are and or5.55and Confirm the roots using a graphing calculator. Let Y1 = x 16x + 64 and Y = 6. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 15] scl: 1 by [ 5, 15] scl: 1 [ 5, 15] scl: 1 by [ 5, 15] scl: 1 Thus,thesolutionsare and. 44.PHYSICALSCIENCE For an experiment in physics class, a water balloon is dropped from the window of the school building. The window is 40 feet high. How long does it take until the balloon hits the ground? Round to the nearest hundredth. SOLUTION: Use the formula h = 16t + h 0 to approximate the number of seconds it takes for the balloon to hit the ground. At ground level, h = 0 and the initial height is 40, so h 0= 40. Page 7
28 [ 5, 15] scl: 1 by [ 5, 15] scl: 1 Squares Thus,thesolutionsare and. 44.PHYSICALSCIENCE For an experiment in physics class, a water balloon is dropped from the window of the school building. The window is 40 feet high. How long does it take until the balloon hits the ground? Round to the nearest hundredth. SOLUTION: Use the formula h = 16t + h 0 to approximate the number of seconds it takes for the balloon to hit the ground. At ground level, h = 0 and the initial height is 40, so h 0= 40. The roots are 1.58and1.58.Thetimetheballoonfallscannotbenegative. So, it takes about 1.58 seconds for the balloon to hit the ground. 45.SCREENS The area A in square feet of a projected picture on a movie screen can be modeled by the equation A = 0.5d, where d represents the distance from a projector to a movie screen. At what distance will the projected picture have an area of 100 square feet? SOLUTION: Use the formula to find d when A = 100. The roots are 0and0.Thedistancecannotbenegative. So, to have an area of 100 square feet, the projected picture will be 0 feet from the projector. 46.GEOMETRY The area of a square is represented by 9x 4x Find the length of each side. SOLUTION: The length of each side of the square is Page 8.
29 The roots are 0and0.Thedistancecannotbenegative. Squares So, to have an area of 100 square feet, the projected picture will be 0 feet from the projector. 46.GEOMETRY The area of a square is represented by 9x 4x Find the length of each side. SOLUTION: The length of each side of the square is. 47.GEOMETRY The area of a square is represented by 16x + 40x + 5. Find the length of each side. SOLUTION: The length of each side of the square is GEOMETRY The volume of a rectangular prism is represented by the expression 8y + 40y + 50y. Find the possible dimensions of the prism if the dimensions are represented by polynomials with integer coefficients. SOLUTION: Write the volume in factor form to determine possible dimensions for the rectangular prism. The GCF for the terms is y, so factor it out first. So, the possible dimensions are y, y + 5, and y POOLS Ichiro wants to buy an aboveground swimming pool for his yard. Model A is 4 inches deep and holds 1750 cubic feet of water. The length of the rectangular pool is 5 feet more than the width. a. What is the surface area of the water? b. What are the dimensions of the pool? c. Model B pool holds twice as much water as Model A. What are some possible dimensions for this pool? d. Model C has length and width that are both twice as long as Model A, but the height is the same. What is the ratio of the volume of Model A to Model C? SOLUTION: Page 9
30 b. What are the dimensions of the pool? Squares c. Model B pool holds twice as much water as Model A. What are some possible dimensions for this pool? d. Model C has length and width that are both twice as long as Model A, but the height is the same. What is the ratio of the volume of Model A to Model C? SOLUTION: a. Let w = the width of the pool and w+5=thelengthofthepool.thereare1inchesinafoot.sothepoolis4 1 or 3.5 feet deep. The roots are 5 and 0. The width cannot be negative, so the width is 0feet and the length is 5 feet. The surface area of the water is 500 square feet. b. The pool is 0 feet wide by 5 feet long by 4 inches deep. c. Sample answer: ModelBpoolholds 1750or3500cubicfeetofwater. V = wh 3500 = w(3.5) 100 = w The length and width of the pool can be any two numbers that have a product of 100. Since 0 50=100,the dimensions of Model B pool could be 0 ft by 50 ft by 4 inches. d.thewidthofmodelcis 0or40feet.ThelengthofModelCis 5or50feet.Thedepthisstill4inches, or 3.5 feet. The ratio of the volume of Model A to Model C is or. Page 30
31 or 3.5 feet. Squares The ratio of the volume of Model A to Model C is or. 50.GEOMETRY Use the rectangular prism. a. Write an expression for the height and width of the prism in terms of the length,. b. Write a polynomial for the volume of the prism in terms of the length. SOLUTION: a. The height is 8 or the length minus 6. Let minus 10. Then, the width is 10. =thelength.then,theheightis 6. The width is 4 or the length b. 51.CCSSPRECISION A zoo has an aquarium shaped like a rectangular prism. It has a volume of 180 cubic feet. The height of the aquarium is 9 feet taller than the width, and the length is 4 feet shorter than the width. What are the dimensions of the aquarium? SOLUTION: a. Write a polynomial to represent the volume. The dimensions are = w 4, h = w + 9, and w. The volume is 180 cubic feet. Substitute 180 for V and get 0 on one side of the equation. esolutions Manual  Powered by Cognero Use factor by grouping to factor the right side of the equation. Page 31
32 Squares 51.CCSSPRECISION A zoo has an aquarium shaped like a rectangular prism. It has a volume of 180 cubic feet. The height of the aquarium is 9 feet taller than the width, and the length is 4 feet shorter than the width. What are the dimensions of the aquarium? SOLUTION: a. Write a polynomial to represent the volume. The dimensions are = w 4, h = w + 9, and w. The volume is 180 cubic feet. Substitute 180 for V and get 0 on one side of the equation. Use factor by grouping to factor the right side of the equation. The value of w can be 6, 6, or 5. However, no measurement of length can be negative, so w = 6, and h = w + 9 or 15. = w 4 or, Therefore, the aquarium is 6 feet wide, feet long, and 15 feet high. 5.ELECTION For the student council elections, Franco is building the voting box shown with a volume of 96 cubic inches. What are the dimensions of the voting box? SOLUTION: esolutions Manual  Powered by Cognero Write a polynomial to represent the volume. Page 3
33 The value of w can be 6, 6, or 5. However, no measurement of length can be negative, so w = 6, and h = w + 9 or 15. Squares Therefore, the aquarium is 6 feet wide, feet long, and 15 feet high. = w 4 or, 5.ELECTION For the student council elections, Franco is building the voting box shown with a volume of 96 cubic inches. What are the dimensions of the voting box? SOLUTION: Write a polynomial to represent the volume. The dimensions are = h + 8, w = h, and h. The volume is 96 cubic inches. Substitute 96 for V and get 0 on one side of the equation. Use factor by grouping to factor the right side of the equation. The value of h can be 4, 4, or 6. However, no measurement of length can be negative, so h = 4, and w = h or. = h + 8 or 1, So, the voting box is 4 inches high, 1 inches long and inches wide ERRORANALYSIS Debbie and Adriano are factoring the expression x x completely. Is either of them correct? Explain your reasoning. Page 33
34 The value of h can be 4, 4, or 6. However, no measurement of length can be negative, so h = 4, and w = h or. Squares So, the voting box is 4 inches high, 1 inches long and inches wide. 8 = h + 8 or 1, 4 53.ERRORANALYSIS Debbie and Adriano are factoring the expression x x completely. Is either of them correct? Explain your reasoning. SOLUTION: Adriano is correct. Debbie did not factor the expression completely. She should have factored x 1 into (x 1)(x + 1). 54.CHALLENGE Factor x n+6 +x n+ n + x completely. SOLUTION: n The greatest common factor of each term is x. x n+6 +x n+ n n 6 + x = x (x + x + 1) 55.OPENENDED Write a perfect square trinomial equation in which the coefficient of the middle term is negative and the last term is a fraction. Solve the equation. SOLUTION: Find a,b and c in the trinomial ax + bx + c, sothatcoefficientofthemiddletermisnegativeandthelasttermisa fraction. Choose a fraction for the last term where the numerator and denominator are perfect squares. Let a = 1. The coefficient of the middle terms is. Then b is 3.Thetrinomialwillthenbe. x 3x + =0 The solution is. 56.REASONING Find a counterexample to the following statement. A polynomial equation of degree three always has three real solutions. Page 34
35 54.CHALLENGE Factor x +x + x completely. SOLUTION: n The greatest common factor of each term is x. Squares n+6 n+ n n 6 x +x + x = x (x + x + 1) 55.OPENENDED Write a perfect square trinomial equation in which the coefficient of the middle term is negative and the last term is a fraction. Solve the equation. SOLUTION: Find a,b and c in the trinomial ax + bx + c, sothatcoefficientofthemiddletermisnegativeandthelasttermisa fraction. Choose a fraction for the last term where the numerator and denominator are perfect squares. Let a = 1. The coefficient of the middle terms is. Then b is 3.Thetrinomialwillthenbe. x 3x + =0 The solution is. 56.REASONING Find a counterexample to the following statement. A polynomial equation of degree three always has three real solutions. SOLUTION: Sample answer: You can create an equation of degree 3 that has only one solution by choosing a linear factor and a quadratic factor that cannot equal 0.For example the factor (x + 1) cannot equal zero for any real value of x. The 3 3 product of the linear factor (x + 1) and (x + 1) is x + x + x+1.thepolynomialequationx + x + x + 1 = 0 only has one solution since only the factor (x + 1) can equal zero when x = 1.Thus, this polynomial equation of degree has only one solution. [Since the only solution for x = 1 is x = 1, another counterexample could be x 1 = 0.] 57.CCSSREGULARITY Explain how to factor a polynomial completely. SOLUTION: First look for a GCF in all the terms and factor the GCF out of all the terms. Then, if the polynomial has two terms, check if the terms are the differences of squares and factor if so. If the polynomial has three terms, check if the trinomial will factor into two binomial factors or if it is a perfect square trinomial and factor if so. If the polynomial has four or more terms, factor by grouping. If the polynomial does not have a GCF and cannot be factored, the polynomial is a prime polynomial. 58.WHICH ONE DOESN TBELONG? Identify the trinomial that does not belong. Explain. SOLUTION: esolutions Manual  Powered by Cognero Identify the first, last and middle terms. Write them as perfect squares if possible. Page 35
36 First look for a GCF in all the terms and factor the GCF out of all the terms. Then, if the polynomial has two terms, check if the terms are the differences of squares and factor if so. If the polynomial has three terms, check if the trinomial will factor into two binomial factors or if it is a perfect square trinomial and factor if so. If the polynomial has foursquares or more terms, factor by grouping. If the polynomial does not have a GCF and cannot be factored, the polynomial is a prime polynomial. 58.WHICH ONE DOESN TBELONG? Identify the trinomial that does not belong. Explain. SOLUTION: Identify the first, last and middle terms. Write them as perfect squares if possible. Trinomial First Term Last Term Middle Terms 4x 36x x = (x) 81 = 9 36x = x 9 5x + 10x + 1 5x = (5x) 1 = 1 10x = 5x 1 4x + 10x + 4 4x = (4x) 4 = 10x 4x = 16x 9x 4x x = (3x) 16 = 4 4x = 3x 4 In each of the polynomials, 4x 36x + 81, 5x + 10x + 1, and 9x 4x + 16, the first and last terms are perfect squares and the middle terms are ab. So, they are perfect square trinomials. The trinomial 4x + 10x + 4 is not a perfect square because the middle term is not ab. 59.OPENENDED Write a binomial that can be factored using the difference of two squares twice. Set your binomial equal to zero and solve the equation. SOLUTION: Find a and b so that the binomial a b that can be factored using the difference of two squares twice. Let b = Then choose a such that it is raised to the 4th power. Let a = x. Then a b = x 1. The solutions are 1 and WRITINGINMATH Explain how to determine whether a trinomial is a perfect square trinomial. SOLUTION: Determine if the first and last terms are perfect squares. Then determine if the middle term is equal to timesthe product of the principal square roots of the first and last terms. If these three criteria are met, the trinomial is a perfect square trinomial. 61.What is the solution set for the equation (x 3) = 5? Manual  Powered by Cognero esolutions A { 8, } Page 36
37 SOLUTION: Determine if the first and last terms are perfect squares. Then determine if the middle term is equal to timesthe product Squares of the principal square roots of the first and last terms. If these three criteria are met, the trinomial is a perfect square trinomial. 61.What is the solution set for the equation (x 3) = 5? A { 8, } B {, 8} C {4, 14} D { 4, 14} SOLUTION: The solutions are and 8, so the correct choice is B. 6.SHORTRESPONSE Write an equation in slopeintercept form for the graph shown. SOLUTION: The line passes through the points (, 0) and (0, 4). Use these points to find the slope. The yintercept is 4. So, the equation of the line in slopeintercept form is y = x At an amphitheater, the price of lawn seats and pavilion seats is $10. The price of 3 lawn seats and 4 pavilion seats is $5. How much do lawn and pavilion seats cost? F $0 and $41.5 G $10 and $50 Page 37
38 The yintercept is 4. Squares So, the equation of the line in slopeintercept form is y = x At an amphitheater, the price of lawn seats and pavilion seats is $10. The price of 3 lawn seats and 4 pavilion seats is $5. How much do lawn and pavilion seats cost? F $0 and $41.5 G $10 and $50 H $15 and $45 J $30 and $30 SOLUTION: Let l = the price of a lawn seat and let p = the price of a pavilion seat. Then, l + p = 10 and 3l + 4p = 5. Use the value of l and either equation to find the value of p. The cost of a lawn seat is $15 and the cost of a pavilion seat is $45. So, the correct choice is H. 64.GEOMETRY The circumference of a circle is units.whatistheareaofthecircle? A B C D SOLUTION: The radius of the circle is units. Page 38
39 Squares The cost of a lawn seat is $15 and the cost of a pavilion seat is $45. So, the correct choice is H. 64.GEOMETRY The circumference of a circle is units.whatistheareaofthecircle? A B C D SOLUTION: The radius of the circle is units. The area of the circle is. So, the correct choice is A. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 65.x 16 SOLUTION: 66.4x 81y SOLUTION: p SOLUTION: 68.3a 0 Page 39
40 SOLUTION: Squares p SOLUTION: 68.3a 0 SOLUTION: The polynomial 3a 0 has no common factors or perfect squares. It is prime. 69.5n 1 SOLUTION: c SOLUTION: Solve each equation. Confirm your answers using a graphing calculator. 71.4x 8x 3 = 0 SOLUTION: The roots are and4. Confirm the roots using a graphing calculator. Let Y1 = 4x 8x 3 and Y = 0. Use the intersect option from the CALCmenutofindthepointsofintersection. Page 40
41 SOLUTION: Squares Solve each equation. Confirm your answers using a graphing calculator. 71.4x 8x 3 = 0 SOLUTION: The roots are and4. Confirm the roots using a graphing calculator. Let Y1 = 4x 8x 3 and Y = 0. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 10, 10] scl: 1 by [ 40, 10] scl: 5 [ 10, 10] scl: 1 by [ 40, 10] scl: 5 Thus, the solutions are and x 48x + 90 = 0 SOLUTION: Page 41
42 [ 10, 10] scl: 1 by [ 40, 10] scl: 5 Thus, thesquares solutions are and x 48x + 90 = 0 SOLUTION: Therootsare3and5. Confirm the roots using a graphing calculator. Let Y1 = 6x 48x + 90 and Y = 0. Use the intersect option from the CALC menu to find the points of intersection. [ 10, 10] scl: 1 by [ 10, 10] scl: 1 [ 10, 10] scl: 1 by [ 10, 10] scl: 1 Thus, the solutions are 3 and x + 14x = 8 SOLUTION: esolutions Manual  Powered by Cognero Page 4
43 [ 10, 10] scl: 1 by [ 10, 10] scl: 1 Squares Thus, the solutions are 3 and x + 14x = 8 SOLUTION: The roots are and1. Confirm the roots using a graphing calculator. Let Y1 = 14x + 14xandY = 8. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 5, 5] scl: 1 by [ 5, 35] scl: 4 [ 5, 5] scl: 1 by [ 5, 35] scl: 4 Thus, the solutions are and x 10x = 48 SOLUTION: Page 43
44 [ 5, 5] scl: 1 by [ 5, 35] scl: 4 Squares Thus, the solutions are and x 10x = 48 SOLUTION: The roots are 3and8. Confirm the roots using a graphing calculator. Let Y1 = x 10x and Y = 48. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 10, 10] scl: 1 by [ 0, 60] scl: 6 [ 10, 10] scl: 1 by [ 0, 60] scl: 6 Thus, the solutions are 3 and x 5x = 30 SOLUTION: Page 44
45 [ 10, 10] scl: 1 by [ 0, 60] scl: 6 Squares Thus, the solutions are 3 and x 5x = 30 SOLUTION: Therootsareand3. Confirm the roots using a graphing calculator. Let Y1 = 5x 5x and Y = 30. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 4, 7] scl: 1 by [ 45, 5] scl: 5 [ 4, 7] scl: 1 by [ 45, 5] scl: 5 Thus, the solutions are and x 16x = 19 SOLUTION: Page 45
46 [ 4, 7] scl: 1 by [ 45, 5] scl: 5 Squares Thus, the solutions are and x 16x = 19 SOLUTION: The roots are 4and6. Confirm the roots using a graphing calculator. Let Y1 = 8x 16x and Y = 19. Use the intersect option from the CALCmenutofindthepointsofintersection. [ 10, 10] scl: 1 by [0, 00] scl: 0 [ 10, 10] scl: 1 by [0, 00] scl: 0 Thus, the solutions are 4 and 6. SOUNDTheintensityofsoundcanbemeasuredinwattspersquaremeter.Thetablegivesthewatts per square meter for some common sounds. Page 46
47 [ 10, 10] scl: 1 by [0, 00] scl: 0 Squares Thus, the solutions are 4 and 6. SOUNDTheintensityofsoundcanbemeasuredinwattspersquaremeter.Thetablegivesthewatts per square meter for some common sounds. 77.How many times more intense is the sound from busy street traffic than sound from normal conversation? SOLUTION: To find how many times more intense the sound from busy street traffic is than sound from normal conversation, divide the watts per square meter for busy street traffic by the watts per square meter for normal conversation. 1 So, the sound from busy street traffic is 10 or 10 times more intense that the sound from normal conversation. 78.Which sound is 10,000 times as loud as a busy street traffic? SOLUTION: To find which sound is 10,000 times as loud as a busy street traffic, multiply the watts per square meter for busy streettrafficby10,000or So, the sound at the front rows of a rock concert is 10,000 times as loud as a busy street traffic. 79.How does the intensity of a whisper compare to that of normal conversation? SOLUTION: To find how the intensity of a whisper compares to that of normal conversation, divide the watts per meter for a whisper by the watts per square meter for a normal conversation. esolutions Manual  Powered by Cognero So, the intensity of a whisper is Page 47 that of a normal conversation.
48 Squares So, the sound at the front rows of a rock concert is 10,000 times as loud as a busy street traffic. 79. How does the intensity of a whisper compare to that of normal conversation? SOLUTION: To find how the intensity of a whisper compares to that of normal conversation, divide the watts per meter for a whisper by the watts per square meter for a normal conversation. So, the intensity of a whisper is that of a normal conversation. Find the slope of the line that passes through each pair of points. 80. (5, 7), (, 3) SOLUTION: 81. (, 1), (5, 3) SOLUTION: 8. ( 4, 1), ( 3, 3) SOLUTION: Page 48
49 Squares 8. ( 4, 1), ( 3, 3) SOLUTION: 83. ( 3, 4), (5, 1) SOLUTION: 84. (, 3), (8, 3) SOLUTION: 85. ( 5, 4), ( 5, 1) SOLUTION: Page 49
88 Differences of Squares. Factor each polynomial. 1. x 9 SOLUTION: 2. 4a 25 SOLUTION: 3. 9m 144 SOLUTION: 4. 2p 162p SOLUTION: 5.
Factor each polynomial. 1.x 9 SOLUTION:.a 5 SOLUTION:.9m 1 SOLUTION:.p 16p SOLUTION: 5.u 81 SOLUTION: Page 1 5.u 81 SOLUTION: 6.d f SOLUTION: 7.0r 5n SOLUTION: 8.56n c SOLUTION: Page 8.56n c SOLUTION:
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